
%!TEX program = xelatex
%!TEX TS-program = xelatex
%!TEX encoding = UTF-8 Unicode

\documentclass[10pt]{article} 

\input{wang_preamble.tex}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{titling}
\setlength{\droptitle}{-2cm}   % This is your set screw

%%文档的题目、作者与日期
\author{UID \underline{\hspace{4cm}} \hspace{0.6cm} NAME \underline{\hspace{4cm}} }
\title{Mathematical Writing Exercise Chapter 02 (2.10-2.14)}
%\date{\vspace{-3ex}}
%\renewcommand{\today}{\number\year \,年 \number\month \,月 \number\day \,日}
%\date{2023 年 10 月 31 日}
%\date{March 9, 2021}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 01
True or False. 
\begin{enumerate}[label={(\arabic*)}]
\item  Parallelism is the expression of parallel ideas in parallel grammatical form, including in mathematical expressions. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)
 
\item  Parallelism should be used, where appropriate, to aid readability and understanding. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)
 
\item  Consider this extract:
\begin{center}
\fbox{
\begin{minipage}{12cm}
The Cayley transform is defined by $C = (A - \theta_1I)^{-1} (A - \theta_2I)$.
If $\lambda$ is an eigenvalue of $A$ then
$$(\lambda-\theta_2)(\lambda-\theta_1)^{-1}$$ 
is an eigenvalue of $C$. 
\end{minipage}
}
\end{center}
% \dotfill (\,\,\,\,\,\,\,\,\,\,)
 
\item  The factors in the eigenvalue expression are presented in the reverse order to the factors in the expression for $C$. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)
 
\item  This may confuse the reader, who might, at first, think there is an error. 
The two expressions should be ordered in the same way.
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 02
True or False. 
\begin{enumerate}[label={(\arabic*)}]
\item  Parallelism works at many levels, from equations and sentences to theorem statements and section headings. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  It should be borne in mind throughout the writing process. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  If one theorem is very similar to another, the statements should reflect that - the wording should not be changed just for the sake of variety. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  However, it is perfectly acceptable to economize on words by saying in Theorem 2, say, ``Under the conditions of Theorem 1.''
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 03
\begin{enumerate}[label={(\arabic*)}]
\item  For a more subtle example, consider the sentence
\begin{center}
\fbox{
\begin{minipage}{12cm}
It is easy to see that $f(x, y) > 0$ for $x > y$. 
\end{minipage}
}
\end{center}

\item  In words, this sentence is read as ``It is easy to see that $f(x,y)$ is greater than zero for $x$ greater than $y$.'' 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  The first > translates to ``is greater than'' and the second to ``greater than,'' so there is a lack of parallelism, which
the reader may find disturbing. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  A simple cure is to rewrite the sentence:
\begin{center}
\fbox{
\begin{minipage}{12cm}
It is easy to see that $f(x, y) > 0$ when $x > y$. \\ 
It is easy to see that if $x > y$ then $f(x, y) > 0$.
\end{minipage}
}
\end{center}
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 04
Glossary for Mathematical Writing. 

\begin{enumerate}[label={(\arabic*)}]
\item  Without loss of generality = \underline{\hspace{0.6cm}}\, \underline{\hspace{0.6cm}}\, \underline{\hspace{0.6cm}}\, \underline{\hspace{0.6cm}}\, \underline{\hspace{0.6cm}}\, \underline{\hspace{0.6cm}}\, \underline{\hspace{0.6cm}}. %I have done an easy special case.
\item  By a straightforward computation = \underline{\hspace{0.6cm}}\, \underline{\hspace{0.6cm}}\, \underline{\hspace{0.6cm}}\, \underline{\hspace{0.6cm}}.%I lost my notes.
\item  The details are left to the reader = \underline{\hspace{0.6cm}}\, \underline{\hspace{0.6cm}}\, \underline{\hspace{0.6cm}}. %I can't do it.
\item  The following alternative proof of X's result may be of interest = \underline{\hspace{0.6cm}}\, \underline{\hspace{0.6cm}}\, \underline{\hspace{0.6cm}}\, \underline{\hspace{0.6cm}}. %I cannot understand X.
\item  It will be observed that = \underline{\hspace{0.6cm}}\, \underline{\hspace{0.6cm}}\, \underline{\hspace{0.6cm}}\, \underline{\hspace{0.6cm}}\, \underline{\hspace{0.6cm}}\, \underline{\hspace{0.6cm}}. %I hope you hadn't noticed that.
\item  Correct to within an order of magnitude = \underline{\hspace{0.6cm}}. %wrong.
\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 05
Good or Bad. 
\begin{enumerate}[label={(\arabic*)}]
\item  Here are two further examples of lack of parallelism that can readily arise in mathematical writing.
\item  The result is obtained by using Lemma 2.1, Theorems 2.5 and 3.1.
\dotfill (\,\,\,\,\, \,\,\,\,\,\,\,\,\,)
\item  The result is obtained by using Lemma 2.1 and Theorems 2.5 and 3.1.
\dotfill (\,\,\,\,\,\,\,\,\,\,\,\,\,\,)
\item  These bounds are sharp, scale independent, but expensive to compute.
\dotfill (\,\,\,\,\,\,\,\,\,\,\,\,\,\,)
\item  These bounds are sharp and scale independent but expensive to compute.
\dotfill (\,\,\,\,\,\,\,\,\,\,\,\,\,\,)
\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 06
True or False. 
\begin{enumerate}[label={(\arabic*)}]
\item  By convention, variables are set in italic font and standard mathematical functions such as sin, cos, arctan, max, gcd, trace, and lim are set in roman font, as are multiple-letter variable names. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  It is a common mistake to set functions in italic, which is ambiguous. Compare $\tan x$ with $tanx$, which looks like the product of four scalars. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  A little-known 2009 ISO standard specifies in detail how all kinds of mathematical notation should be typeset. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  The most interesting aspects of the standard concern the use of roman versus italic fonts. 
The standard requires mathematical constants whose values do not change to be written in roman. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Thus we should write e for the base of the natural logarithm and i for the imaginary unit. However, standard \LaTeX 
fonts do not have upright lowercase Greek letters, so an italic $\pi$ is unavoidable. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  The standard also requires mathematical operators to be written in roman, including the ``d'' in derivatives and integrals. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Thus we should write
$$\frac{\mathrm{d}}{\mathrm{d}x}f(x), \hspace{1cm} \int_0^1 f(x)\mathrm{d}x,  \hspace{1cm} \int_C \frac{e^z}{z}\mathrm{d}z=2\pi \mathrm{i} $$
instead of
$$\frac{d}{dx}f(x),  \hspace{1cm} \int_0^1 f(x)dx,  \hspace{1cm} \int_C \frac{e^z}{z}dz=2\pi i.  $$
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  These uses of roman fonts are by no means universal, but they are appealing in that they distinguish variable quantities from fixed ones. 
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  While using these conventions in editing {\it The Princeton Companion to Applied Mathematics}, I grew to like them and now use them in most of my writing.
 \dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 07
True or False. 
\begin{enumerate}[label={(\arabic*)}]
\item  In mathematical writing we often use an ellipsis (three dots) to denote omission in an expression. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  There are two different kinds of ellipsis: vertically centred $( \cdots )$ and ``ground level'' or ``baseline'' $( ... )$. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Vertically centered dots are used between operators that sit above the baseline, such as $+$, $-$, $=$, and $<$. 
Ground level dots are used in a list or to indicate a product. 
Examples:
$$ x_1 + x_2 + \cdots + x_n, \,\,\, \sigma_1 > \sigma_2 > \cdots > \sigma_n,\,\,\,\lambda_1, \lambda_2, ..., \lambda_n, \,\,\, A_1A_2...A_n.$$
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  An operator or comma should be symmetrically placed around the ellipsis; thus $x_1 + x_2 + \cdots x_n$ and $\lambda_1, \lambda_2, ... \lambda_n$ are correct. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  When an ellipsis falls at the end of a sentence there is the question of how the period (a ``full stop'' in British English) is treated. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Recommendations vary. 
{\it The Chicago Manual of Style} suggests typing the period before the three ellipsis points (so that there is no space between the first of the four dots and the preceding character). 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  When the ellipsis is part of a mathematical formula it seems natural to put it before the period, but the two possibilities may be visually indistinguishable, as in the sentence 
\begin{center}
\fbox{
\begin{minipage}{12cm}
The Mandelbrot set is defined in terms of the iteration $z_{k+1} =z_k^2 + c, k = 0,1,2,...$.
\end{minipage}
}
\end{center}
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 08
True or False. (Avoid Unnecessary Symbols.)
\begin{enumerate}[label={(\arabic*)}]
\item  Do not use mathematical symbols unless they serve a purpose. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  In the sentence ``A symmetric positive definite matrix $A$ has real eigenvalues'' there is no need to name the matrix unless the name is used in a subsequent sentence. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Similarly, in the sentence ``This algorithm has $t =\log_2 n$ stages,'' the ``$t =$'' can be omitted unless $t$ is defined in this sentence and used immediately. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Watch out for unnecessary parentheses, as in the phrase ``the matrix $(A - \lambda I)$ is singular.'' 
... \dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 9
Good or Bad. (Fine-tune Theorem Statements.) 
\begin{enumerate}[label={(\arabic*)}]
\item  Having obtained a theorem, don't neglect to fine-tune the statement. 
Avoid automatically beginning every theorem with a let clause. 

\item  Theorem. Let $f(x)$ be continuous on $[a,b]$ and differentiable on $(a, b)$. 
Then $$ f(b) - f(a) = (b-a)f'(\xi) \,\,\mathrm{for\,\, some}\,\,  \xi\in (a, b).$$ 
\dotfill (\,\,\,\,\,\,\,\,\,\,\,\,\,\,)

\item  Theorem. If $f(x)$ is continuous on $[a,b]$ and differentiable on $(a, b)$ then
$$ f(b) - f(a) = (b-a)f'(\xi) \,\,\mathrm{for\,\, some}\,\,  \xi\in (a, b).$$ 
\dotfill (\,\,\,\,\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 10
Good, Fair or Bad. (Place Symbols Carefully) 
\begin{enumerate}[label={(\arabic*)}]
\item  Avoid starting a sentence with a mathematical expression, particularly if a previous sentence ended with one, otherwise the reader may have difficulty parsing the sentence. 

\item  For example, ``$A$ is an ill-conditioned matrix'' (possible confusion with the word ``A'') can be changed to ``The matrix $A$ is ill conditioned.'' 

\item  Separate mathematical symbols by punctuation marks or words, if possible, for the same reason. 

\item  If $x > 1$ $f(x) < 0$.
\dotfill (\,\,\,\,\,\,\,\,\,\,\,\,\,\,)

\item  If $x> 1$, $f(x) < 0$.
\dotfill (\,\,\,\,\,\,\,\,\,\,\,\,\,\,)

\item  If $x> 1$ then $f(x) < 0$.
\dotfill (\,\,\,\,\,\,\,\,\,\,\,\,\,\,)

\item  Since $p^{-1} + q^{-1} = 1$, $\lVert \cdot \rVert_p$ and $\lVert \cdot \rVert_q$ are dual norms. 
\dotfill (\,\,\,\,\,\,\,\,\,\,\,\,\,\,)

\item  Since $p^{-1} + q^{-1} = 1$, the norms $\lVert \cdot \rVert_p$ and $\lVert \cdot \rVert_q$ are dual.
\dotfill (\,\,\,\,\,\,\,\,\,\,\,\,\,\,)

\item  It suffices to show that $\lVert H \rVert_p = n^{1/p}, 1 \le p \le 2$.
\dotfill (\,\,\,\,\,\,\,\,\,\,\,\,\,\,)

\item  It suffices to show that $\lVert H \rVert_p = n^{1/p}, (1 \le p \le 2$).
\dotfill (\,\,\,\,\,\,\,\,\,\,\,\,\,\,)

\item  It suffices to show that $\lVert H \rVert_p = n^{1/p}$ for $1 \le p \le 2$.
\dotfill (\,\,\,\,\,\,\,\,\,\,\,\,\,\,)

\item  For $n = r$ (2.2) holds with $\delta_r = 0$.
\dotfill (\,\,\,\,\,\,\,\,\,\,\,\,\,\,)

\item  For $n = r$, (2.2) holds with $\delta_r = 0$.
\dotfill (\,\,\,\,\,\,\,\,\,\,\,\,\,\,)

\item  For $n = r$, inequality (2.2) holds with $\delta_r = 0$.
\dotfill (\,\,\,\,\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 11
Good, Fair or Bad. (Choose between ``The'' or ``A'') 
\begin{enumerate}[label={(\arabic*)}]
\item  In mathematical writing the use of the article ``the'' can be inappropriate when the object to which it refers is (potentially) not unique or does not exist. 

\item  Rewording, or changing the article to ``a,'' usually solves the problem. 

\item  Let the Schur decomposition of $A$ be $QTQ^*$.
\dotfill (\,\,\,\,\,\,\,\,\,\,\,\,\,\,)

\item  Let a Schur decomposition of $A$ be $QTQ^*$.
\dotfill (\,\,\,\,\,\,\,\,\,\,\,\,\,\,)

\item  Let $A$ have the Schur decomposition $QTQ^*$.
\dotfill (\,\,\,\,\,\,\,\,\,\,\,\,\,\,)

\item  Let $A = QTQ^*$ be a Schur decomposition. 
\dotfill (\,\,\,\,\,\,\,\,\,\,\,\,\,\,)

\item  Under what conditions does the iteration converge to the solution of $f(x) = 0$?
\dotfill (\,\,\,\,\,\,\,\,\,\,\,\,\,\,)

\item  Under what conditions does the iteration converge to a solution of $f(x) = 0$?
\dotfill (\,\,\,\,\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 12
Rewrite. (Look for Notational Synonyms) 
\begin{enumerate}[label={(\arabic*)}]
\item  Sometimes you have a choice of notational synonyms, one of which is preferable. 

\item  Rewrite the following examples in a way that is more aesthetically pleasing or easier to read (a capital letter denotes a matrix).

\item  $\sqrt{\sum\limits_{i,j} (a_{i,j}-b_{i,j})^2}$. 
\dotfill (\hspace{2cm})

\item  $e^{\frac{2\pi i}{\sqrt{x^2+y^2}}}$. 
\dotfill (\hspace{2cm})

\item  $\frac{\lvert L\rvert  \lvert U\rvert }{1-n\epsilon}$. 
\dotfill (\hspace{2cm})

\item  $X_{k+1} = \frac{X_k}{2}[3I-X_k^2]$. 
\dotfill (\hspace{2cm})

\item  $\min\{\epsilon \mid \lvert b-Ay \rvert \le \epsilon \lvert A\rvert \lvert y\rvert \}$. 
\dotfill (\hspace{2cm})

\item  $\underset{P\, a\, permutation} {\underset{U^TU=I}{\min} } \lVert A-UPB\rVert$. 
\dotfill (\hspace{2cm})

\item  $x=\begin{bmatrix}x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$. 
\dotfill (\hspace{2cm})

\item  $\Lambda = \begin{bmatrix}\lambda_1&&& \\ & \lambda_2 && \\ &&\ddots& \vdots \\ &&&\lambda_n \end{bmatrix}$. 
\dotfill (\hspace{2cm})

\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 13
True or False. (Expand Equation References) 
\begin{enumerate}[label={(\arabic*)}]
\item   When you reference an earlier equation it helps the reader if you add a word or phrase describing the nature of that equation. 
The aim is to save the reader the trouble of turning back to look at the earlier equation. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  For example, ``From the definition (6.2) of dual norm'' is more helpful than ``From (6.2)''; and ``Combining the recurrence (3.14) with inequality (2.9)'' is more helpful than ``Combining (3.14) and (2.9).'' 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  David Mermin calls this advice the ``Good Samaritan Rule.'' 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  As in these examples, the word added should be something more informative than just ``equation'' (or the ugly abbreviation ``Eq.''), and inequalities, implications, and lone expressions should not be referred to as equations.
\dotfill (\,\,\,\,\,\,\,\,\,\,)


\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 14
True or False. (Miscellaneous) 
\begin{enumerate}[label={(\arabic*)}]
\item  When you are using i as the imaginary unit it is also best to use $i$ as a counting index, to add confusion, even if (as here) roman and italic fonts distinguish the two. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  More generally, do not use a letter as a dummy variable if it is already being used for another purpose. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Note the difference between the Greek letter epsilon, $\epsilon$, and the ``belongs to'' symbol $\in$, as in $\lVert x\rVert < \epsilon$ and $x \in \mathbb{R}^n$. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Another version of the Greek epsilon is $\varepsilon$; the two variants are typed in \TeX \, as \verb+\epsilon+ ($\epsilon$) and \verb+\varepsilon+ ($\varepsilon$). 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Note the distinction between the Greek letter $\Pi$ (uppercase) or $\pi$ (lowercase) and the product symbol $\prod$. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 15
True or False. (Miscellaneous) 
\begin{enumerate}[label={(\arabic*)}]

\item  In bracketing multilayered expressions you have a choice of fences for the layers and a choice of sizes, for example \verb+{[(+, this ordering being the one recommended by {\it The Chicago Manual of Style}. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Most authors try to avoid mixing different fences in the same expression, as it leads to a rather muddled appearance. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Write ``the $k$th term'', not ``the $k^{\mathrm{th}}$ term'',  ``the $k$'th term'', or ``the $k$-th term''. (It is interesting to note that nth is a genuine word that can be found in most dictionaries.) 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  A slashed exponent, as in $y^{1/2}$, is generally preferable to a stacked one, as in $y^{\frac{1}{2}}$. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 16
True or False. (Miscellaneous) 
\begin{enumerate}[label={(\arabic*)}]

\item  The standard way to express that $i$ is to take the values $1$ to $n$ in steps of 1 is to write
$$ i = 1, ..., n \,\, \mathrm{or}\,\, i = 1, 2, ..., n, $$ 
where all the commas are required. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  An alternative notation originating in programming languages such as Fortran 90 and MATLAB is $i = 1: n$. 
For counting down we can write $i = n, n-1, ..., 1$ or $i = n: -1: 1$, where the middle integer denotes the increment. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  This notation is particularly convenient when extended to describe submatrices: $A(i:j, p:q)$ denotes the submatrix formed from the intersection of rows $i$ to $j$ and columns $p$ to $q$ of the matrix $A$. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)


\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 17
True or False. (Miscellaneous) 
\begin{enumerate}[label={(\arabic*)}]

\item  Avoid (or rewrite) tall in-line expressions, which can disrupt the line spacing. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  For example, the vector $\begin{bmatrix} g_1 \\ g_2 \end{bmatrix}$ can be rewritten as $[g_1^T, g_2^T]^T$. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  A vertically centered dot is useful for denoting multiplication in expressions where terms need to be separated for clarity:
\begin{eqnarray*}
16046641 &=& 13\cdot 37\cdot 73 \cdot 457, \\ 
\mathrm{cond}(A,x) &=& \frac{ \lVert \lvert I-A^+A \rvert \cdot \lvert A^T\rvert \cdot \lvert {A^+}^Tx\rvert \rVert }{\lVert x \rVert}. 
\end{eqnarray*}
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 18
True or False. (Miscellaneous) 
\begin{enumerate}[label={(\arabic*)}]
\item  Care is needed to add ambiguity in slashed fractions. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  For example, the expression $-(b-a)^3/12f''(\eta)$ is better written as $-((b-a)^3/12)f''(\eta)$ or $-f''(\eta)(b-a)^3/12$. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 19
Good or Bad. (Miscellaneous) 
\begin{enumerate}[label={(\arabic*)}]
\item  Care is needed to ensure that a footnote symbol does not appear to be part of an equation. 

\item  It can be shown that $\lvert e \rvert < 2u^3$.
\dotfill (\hspace{1cm})

\item  It can be shown that $\lvert e \rvert < 2u$. \footnote{The constant 2 is certainly not optimal. }
\dotfill (\hspace{1cm})

\item  It is a general rule that a footnote mark is placed after punctuation rather than before it. 

\item  If there is no punctuation it is better to rewrite so that the footnote can be placed on a word. 

\item  The flop count is $8n^3/3$\footnote{Assuming an LU factorization is computed. } for an $n\times n$ matrix. 
\dotfill (\hspace{1cm})

\item  The cost is $8n^3/3$ flops\footnote{Assuming an LU factorization is computed. } for an $n\times n$ matrix. 
\dotfill (\hspace{1cm})

\item  The cost for an $n\times n$ matrix is $8n^3/3$ flops. \footnote{Assuming an LU factorization is computed. }
\dotfill (\hspace{1cm})


\end{enumerate}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\end{enumerate}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


